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Multiplied Prices

by Andrew
(New York)

A customer brought four items to the cashier of a convenience store. The clerk rang up the items and said, “Okay, I multiplied the prices together and your total cost is $7.11.”“You multiplied them?” the customer asked. “You’re supposed to add them, you know.”“I know,” said the clerk, “but it doesn’t make any difference. The total is still $7.11.”What were the prices of the four items?

Comments for Multiplied Prices

A customer brought four items to the cashier of a convenience store. The clerk rang up the items and said, “Okay, I multiplied the prices together and your total cost is $7.11.”
“You multiplied them?” the customer asked. “You’re supposed to add them, you know.”
“I know,” said the clerk, “but it doesn’t make any difference. The total is still $7.11.”
What were the prices of the four items?

Answer:

Solve the problem using integers (cents rather than dollars and cents)

N₁ + N₂ + N₃ + N₄ = 7.11

N₁ * N₂ * N₃ * N₄ = 7.11

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Solve the problem using integers (cents, rather than dollars and cents)

N₁ + N₂ + N₃ + N₄ = 7.11

100N₁ + 100N₂ + 100N₃ + 100N₄ = 100*7.11

100N₁ + 100N₂ + 100N₃ + 100N₄ = 711

N₁ * N₂ * N₃ * N₄ = 7.11

100N₁ * 100N₂ * 100N₃ * 100N₄ = 10⁸*7.11

100N₁ * 100N₂ * 100N₃ * 100N₄ = 711,000,000

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The two equations:

100N₁ + 100N₂ + 100N₃ + 100N₄ = 711

100N₁ * 100N₂ * 100N₃ * 100N₄ = 711,000,000

Or

a + b + c + d = 711

a * b * c * d = 711,000,000

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711000000 = 2⁶ * 3² * 5⁶ * (1*79)

One of the numbers (“a”) must be a multiple of 79, which is a factor of 711000000.

When multiplying b*c*d, the maximum product occurs when all three numbers are equal (b=c=d). Because of this restriction, the product of b*v*c occurs when [(b+c+d)/3]³, or (b+c+d)³/27. This rules out a =395, a = 474, a= 553, a = 632, and a = 711. Calculations are shown below.